Mechanism Design With Money Christos Tzamos S CHOOL OF E LECTRICAL AND C OMPUTER E NGINEERING N ATIONAL T ECHNICAL U NIVERSITY OF A THENS , G REECE
Motivation Introduction Game Theory VCG Mechanisms Mechanism Design Combinatorial Auctions Impossibility Result Motivation Mechanism Design and Social Choice Design rules in order to make decisions based on people preferences when their interests are conflicting. Christos Tzamos Mechanism Design With Money
Motivation Introduction Game Theory VCG Mechanisms Mechanism Design Combinatorial Auctions Impossibility Result Game Theory Game Theory Studies strategic situations where players choose different actions in an attempt to maximize their returns. Outcome Prediction - Solution Concepts Nash Equilibrium Pure Nash Equilibrium Dominant Strategy Christos Tzamos Mechanism Design With Money
Motivation Introduction Game Theory VCG Mechanisms Mechanism Design Combinatorial Auctions Impossibility Result Mechanism Design Mechanism Design Mechanism design is the art of designing rules of a game to achieve a specific outcome under a certain solution concept. Social Choice as a Game A set A of different alternatives A set of n voters (the agents) N Each agent i has a linear order ≻ i ∈ L over the set A A function (mechanism) f : L n → A that maps the agents’ preferences to a single alternative is called social choice function. Christos Tzamos Mechanism Design With Money
Motivation Introduction Game Theory VCG Mechanisms Mechanism Design Combinatorial Auctions Impossibility Result Properties Onto ∀ a ∈ A , ∃ x ∈ L n such that f ( x ) = a Unanimous if ∃ a ∈ A such that ∀ b ∈ A and i ∈ N , a ≻ i b then f ( ≻ 1 , . . . , ≻ n ) = a Pareto Optimal if f ( ≻ 1 , . . . , ≻ n ) = a , then ∄ b ∈ A such that b ≻ i a , ∀ i ∈ N Christos Tzamos Mechanism Design With Money
Motivation Introduction Game Theory VCG Mechanisms Mechanism Design Combinatorial Auctions Impossibility Result Properties - Incentive Compatibility Strategic Manipulation by agent i ∃ ≻ 1 , . . . , ≻ n , ≻ ′ i ∈ L such that b ≻ i a where a = f ( ≻ 1 , . . . , ≻ i , . . . , ≻ n ) and b = f ( ≻ 1 , . . . , ≻ ′ i , . . . , ≻ n ) . Strategyproofness A social choice function is called incentive compatible or strategyproof or truthful if no agent can strategically manipulate it. Christos Tzamos Mechanism Design With Money
Motivation Introduction Game Theory VCG Mechanisms Mechanism Design Combinatorial Auctions Impossibility Result Impossibility Result Gibbard-Satterthwaite Let f be an incentive compatible social choice function onto A , where | A | ≥ 3, then f is a dictatorship. Escape Routes Money Randomization Restricted domain of preferences Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG Setting and Outline Measuring Preferences with Money Each agent has a value for every alternative a ∈ A , which is given by a private function v i : A → R where v i ∈ V i . The value v i ( a ) corresponds to the amount of money agent i is willing to pay in order to force the outcome a . Extending the notion of a mechanism A mechanism is a social choice function f : V 1 × V 2 × · · · × V n and a vector of payment functions p 1 , p 2 , . . . , p n , where p i : V i → R is the amount that player i pays. Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG Desirable Properties Incentive Compatibility Mechanism ( f , p 1 , ..., p n ) is incentive compatible if for each player i , every v 1 , ..., v n and every v ′ i , we have that v − i ) ≥ v i ( f ( v ′ v − i )) − p i ( v ′ v i ( f ( v i ,� v − i )) − p i ( v i ,� i ,� i ,� v − i ) Individual Rationality Mechanism ( f , p 1 , . . . , p n ) is individually rational if for each player i and every v 1 , ..., v n , we have that v i ( f ( v 1 , ..., v n )) − p i ( v i , ..., v n )) ≥ 0 No Positive Transfers Mechanism ( f , p 1 , ..., p n ) has no positive transfers if for each player i and every v 1 , ..., v n , we have that p i ( v i , ..., v n )) ≥ 0 Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG Single Item Auction Selling a single item The set of alternatives is the set of possible winners A = { i − wins | i ∈ N } The agent valuations are v i ( i − wins ) = w i and v i ( j − wins ) = 0 ∀ j � = i . The agent with the highest w i should get the item. How to choose payments No payments Player i should increase his bid to get the item. Winner pays bid In the case where the other agent valuations are lower the winner should decrease his bid Not incentive compatible Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG Single Item Auction - Truthful Mechanisms Pay all agents Give the item to the highest bidder and pay everyone else the value of the winning bid. Truthful but not efficient (positive transfers) Winner pays second highest bid - Vickrey Auction Give the item to the highest bidder i and make him pay p ∗ = max j � = i w j . Everyone else doesn’t pay anything. Truthful and efficient (positive transfers) Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG General Setting Maximizing Social Welfare There exists payment functions such that the mechanism that maximizes the social welfare i.e. � i ∈ N v i ( a ) is incentive compatible. Vickrey-Clarke-Groves Mechanisms f ( v 1 , ..., v n ) ∈ argmax a ∈ A � i ∈ N v i ( a ) p i ( v i ,� v − i ) = h i ( � v − i ) − � j � = i v j ( f ( v 1 , ..., v n )) where h i ∈ V − i → R arbitrary functions Theorem Every VCG mechanism is incentive compatible. Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG Choosing the h i ’s h i = 0 Extremely inefficient. The mechanism pays every player. The no positive transfers property is violated. Clarke Pivot Rule Choose each h i ( � v − i ) = max b ∈ A � j � = i v j ( b ) . (=max social welfare when player i doesn’t participate.) Player i is charged with the amount that the social welfare of the others decreased due to his choice. Theorem A VCG mechanism with the CPR makes no positive transfers. Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG Example Summary & Example a b c price utility I 1 5 10 5 10-5=5 II 7 6 4 0 4 III 9 8 7 0 7 sum 17 19 21 smaller What if player I lies and says that her value for c is 7? than her a b c price utility utility when I 1 5 10 7 2 5-2=3 telling II 7 6 4 the III 9 8 7 truth! sum 17 19 18 Christos Tzamos Mechanism Design With Money
Setting and Outline Introduction Single Item Auction VCG Mechanisms General Settings Combinatorial Auctions Weighted VCG Weighted VCG Affine Maximizer A social choice function f is called affine maximizer if for some player weights w 1 , ..., w n ∈ R and some weights c a ∈ R for all a ∈ A , we have that f ( v 1 , ..., v n ) = argmax a ∈ A ( c a + � i ∈ N w i v i ( a )) Weighted VCG f ( v 1 , ..., v n ) = argmax a ∈ A ( c a + � i ∈ N w i v i ( a )) p i ( v i ,� v − i ) = h i ( � v − i ) − � j � = i ( w j / w i ) v j ( f ( v 1 , ..., v n )) − c a / w i where h i ∈ V − i → R arbitrary functions Theorem (Roberts) If | A | ≥ 3, f is onto A , V i = R | A | , then all incentive compatible mechanisms are Weighted VCG. Christos Tzamos Mechanism Design With Money
Introduction Setting and outline VCG Mechanisms Single-Minded Case Combinatorial Auctions Setting and outline Problem Statement A set of m items M to be divided among the agents ( A = ( N ∪ { e } ) m ) For each agent i, v i : 2 M → R (monotone valuation function) Objective: Maximize social welfare Difficulties The allocation problem is NP-complete to compute optimally How to retrieve and represent agent valuations? ( | A | is exponential) Christos Tzamos Mechanism Design With Money
Introduction Setting and outline VCG Mechanisms Single-Minded Case Combinatorial Auctions Setting and outline Avoiding the difficulties Focus on simpler cases of valuation functions (linear, single minded) Introduce approximation VCG mechanisms VCG mechanisms require the computation of the optimal social welfare. They don’t work for approximations. Christos Tzamos Mechanism Design With Money
Recommend
More recommend
